If the magnitudes of vectors $\overrightarrow{\mathrm{A}},\overrightarrow{\mathrm{B}} \mathrm{and} \overrightarrow{\mathrm{C}}$ and 12,5 and 13 units respectively and $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{C}}$, the angle between vectors  $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ is:

$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{C}}\Rightarrow \left|\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}\right|=\left|\overrightarrow{\mathrm{C}}\right|$
Let angle between $\overrightarrow{\mathrm{A}}$and $\overrightarrow{\mathrm{B}}$ be $\mathrm{\theta }$ then A² + B² + 2ABcos$\mathrm{\theta }$ = $\mathrm{\pi }$/2$\Rightarrow$
$\Rightarrow$ 12² + 5² + 2(12)(5)cos$\mathrm{\theta }$ = 13²
$\Rightarrow$cos$\mathrm{\theta }$ = 0$\Rightarrow$$\mathrm{\theta }$=$\frac{\mathrm{\pi }}{2}\mathrm{rad}$
OR
$\because$13² =12² + 5² i.e.,C² = A² + B²
So,$\overrightarrow{\mathrm{A}}\perp \overrightarrow{\mathrm{B}}$